As others have alluded to, you will require a checkout flight for Big Bear or Catalina. Most rental agreements I came across preclude operations on anything but tarmac. They can also mandate that you are high enough to make landfall when over the water - more applicable to the Catalina crossing. I suspect an instructor would also want you to demonstrate the LAX corridor before you went solo through it. My agreement was 2000 feet, tarmac, minimum. I did Furnace Creek in a C150 in July time; 47 deg C (roughly 6000ft DA if I recall correctly), but Big Bear at 9000ft DA without a turbo would be asking for trouble!
Beware that 'cheap' and 'quality' can be mutually exclusive.